SIAM ReviewPub Date : 2023-08-08DOI: 10.1137/23n975752
Hélène Frankowska
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/23n975752","DOIUrl":"https://doi.org/10.1137/23n975752","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023. <br/> In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical systems theory. In this framework the structural stability theorem says that a $C^1$ dynamical system $dot x =f(x)$ on $mathbb{S} ^1$ is structurally stable if and only if it has finitely many equilibrium points, all of which are hyperbolic. In the above $mathbb{S} ^1$ denotes the unit circle in $mathbb{R}^2$ and a point $x_star$ is called hyperbolic if $f'(x_star) neq 0$. The Peixto density result says that the set of all $C^1$ structurally stable systems on $mathbb{S}^1$ is open and dense in the space of all $C^1$ dynamical systems on $mathbb{S} ^1$ endowed with the $C^1$ norm. The original Peixoto's theorem is more complex and is valid for any smooth closed surface. Its proof, however, is long and not accessible using the tools available to advanced undergraduates, in contrast with the proposed one-dimensional proof, which an undergraduate could follow. This does not mean that the proof itself is elementary. Preliminaries recall all the basic definitions that are needed to successfully conduct the task. The style is rigorous and self-contained. The article also provides some historical comments, making the reading lively and encouraging further learning. The second paper, “Piecewise Smooth Models of Pumping a Child's Swing,” is presented by Brigid Murphy and Paul Glendinning. It concerns models of a child, in either a seated or standing position, swinging on a playground swing. In the article, which arose from the MSc dissertation by one of the authors, these models are analyzed using Lagrangian mechanics and may serve as an introduction to the different ways in which piecewise smooth systems do arise in modeling. The authors describe control strategies of swingers, and, in particular, whether it is possible for the swing to go through a full turn over its pivot. Piecewise smooth terms do naturally appear while discussing the strategies, and this future is analyzed in detail. Indeed the instantaneous reposition of the body of the swinger leads to a jump in the configuration of the swing. Numerical simulations are performed with a standard software package. These investigations would be suitable for undergraduate projects related to classical mechanics courses. At a higher degree level, projects could include further refinement of the existing methods and/or getting more accurate numerical solutions using available specialized software packages. The final section also discusses various related mathematical questions that would be interesting to investigate in this context and mentions other models involving jumps described using piecewise smoot","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 9","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2023-08-08DOI: 10.1137/23m1556435
Jonas Latz
{"title":"Bayesian Inverse Problems Are Usually Well-Posed","authors":"Jonas Latz","doi":"10.1137/23m1556435","DOIUrl":"https://doi.org/10.1137/23m1556435","url":null,"abstract":"SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023. <br/> Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451--559] has given assumptions under which the posterior measure---the Bayesian inverse problem's solution---exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model---the model can be treated as a black box.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"31 10","pages":""},"PeriodicalIF":10.2,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/21m1436658
P. Leenheer, Jack W. Musgrove, Tyler Schimleck
{"title":"A Comprehensive Proof of Bertrand's Theorem","authors":"P. Leenheer, Jack W. Musgrove, Tyler Schimleck","doi":"10.1137/21m1436658","DOIUrl":"https://doi.org/10.1137/21m1436658","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"563-588"},"PeriodicalIF":10.2,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73066658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/23n975703
Hélène Frankowska
{"title":"Education","authors":"Hélène Frankowska","doi":"10.1137/23n975703","DOIUrl":"https://doi.org/10.1137/23n975703","url":null,"abstract":"The Education section in this issue presents two contributions. In `\"Nesterov's Method for Convex Optimization,\" Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, \"A Comprehensive Proof of Bertrand's Theorem,\" is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/23n975673
Marlis Hochbruck
{"title":"Survey and Review","authors":"Marlis Hochbruck","doi":"10.1137/23n975673","DOIUrl":"https://doi.org/10.1137/23n975673","url":null,"abstract":"A point process is called self-exciting if the arrival of an event increases the probability of similar events for some period of time. Typical examples include earthquakes, which frequently cause aftershocks due to increased geological tension in their region; raised intrusion rates in the vicinity of a burglary; retweets in social media incited by some provocative posting; or trading frenzies following a huge stock order. A Hawkes process is a point process that models self-excitement among time events. In contrast to a Markov chain (in which the probability of each event depends only on the state attained in the previous event), chances of arrival of events are increased for some time period after the initial arrival in a Hawkes process. The first Survey and Review paper in this issue, “Hawkes Processes Modeling, Inference, and Control: An Overview,” by Rafael Lima, discusses recent advances in Hawkes process modeling and inference. The parametric, nonparametric, deep learning, and reinforcement learning approaches are covered. Current research challenges for the topic and the real-world limitations of each approach are also addressed. The paper should be of interest to experts in the field, but it also aims to be suitable for newcomers. The second Survey and Review paper, “Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists,” by Laurent Condat, Daichi Kitahara, Andrés Contreras, and Akira Hirabayashi, is dedicated to the solution of convex nonsmooth optimization problems in high-dimensional spaces. The objective function $f$ is assumed to be a sum of simple convex functions $f_j$ with the property that the minimization problem for each $f_j$ is simple, but for $f$ it is hard. For nonsmooth functions, gradient-based optimization algorithms are infeasible. In proximal algorithms, the gradient is replaced by the so-called proximity operator. While closed forms of proximity operators are known for many functions of practical interest, there is no general closed form for the proximity operator of a sum of functions. Therefore, splitting algorithms handle the proximity operators of the functions $f_j$ individually. The paper provides a constructive and self-contained introduction to the class of proximal splitting algorithms. New variants of the algorithms under consideration are developed. Existing convergence results are revisited, unified, and, in some cases, improved. Reading the paper will be rewarding for anyone interested in high-dimensional nonsmooth convex optimization.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2023-05-01DOI: 10.1137/23n975697
None The Editors
{"title":"SIGEST","authors":"None The Editors","doi":"10.1137/23n975697","DOIUrl":"https://doi.org/10.1137/23n975697","url":null,"abstract":"The SIGEST article in this issue is “Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors,” by Antoine Gautier, Francesco Tudisco, and Matthias Hein. Most computational and applied mathematicians will be aware of the results that Perron published in 1907 about the eigensystems of positive matrices, which were then extended by Frobenius in 1912 to the case of nonnegative matrices. This theory has impacted many areas of mathematics, including graph theory, Markov chains, and matrix computation, and it forms a fundamental component in the analysis of a range of models in areas such as demography, economics, wireless networking, and search engine optimization. Our SIGEST article, which first appeared in SIAM Journal on Matrix Analysis and Applications in 2019, extends Perron--Frobenius theory in two directions. First, the authors generalize from matrices to multidimensional arrays. This ties in with one of SIAM Review's most highly cited offerings: •Tensor decompositions and applications, T. G. Kolda and B. W. Bader, SIAM Review, 51 (3) (2009), pp. 455--500. It may also be viewed as extending the theory from graphs to hypergraphs---objects that are currently of much interest, as evidenced by several recent SIAM Review articles, including •Hypergraph cuts with general splitting functions, N. Veldt, A. R. Benson and J. Kleinberg, SIAM Review, 64 (3) (2022), pp. 650--685. By studying this higher-order setting, the authors open up new applications in network science, computer vision, and machine learning. The second major direction of the article is to develop and study nonlinear versions of the underlying spectral problems, and corresponding extensions of the traditional power method. This makes available new classes of iterations for which a comprehensive and satisfactory convergence theory is available. In preparing this SIGEST version, the authors have included new material. The introduction has been extended, and section 2 has been added to provide nontrivial examples of tensor eigenvalue problems in applications, including problems from computer vision and optimal transport. Moreover, subsection 4.1 includes a new nonlinear Perron--Frobenius theorem (Theorem 4.4) that builds on the previously known results in Theorems 4.2. and 4.3.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"123 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
